9.4 Phenomenological Constraints and Implications
9.4.1 Lepton Flavor Mixing Contributions
The contributions to gSRR from δµ(b.1,2) depend on the products of stermion mixing matri- cesZν1m∗Zν2m andZL5iZL4i∗, which are non-vanishing only in the presence of flavor mixing among the first two generations of sneutrinos and RH charged sleptons, respectively. The existence of such flavor mixing also gives rise to lepton flavor violating (LFV) processes such as µ → eγ and µ → e conversion and, indeed, the products Zν1m∗Zν2m andZL5iZL4i∗
enter the rates for such processes at one-loop order. Consequently, the non-observation of LFV processes leads to stringent constraints on these products of mixing matrix elements.
To estimate the order of magnitude for these constraints, we focus on the rate for the decay µ → eγ, which turns out to be particularly stringent. In principle, one could also analyze the constraints implied by limits on theµ → econversion andµ →3ebranching ratios. This would possibly make our conclusions on the maximal size of the flavor violat- ing contributions togRRS more severe, but it would not change our main conclusion: lepton flavor mixing contributions togSRRare unobservably small.
Experimentally, the most stringent bound on the corresponding branching ratio has been obtained by the MEGA collaboration[116]:
Br(µ→eγ)≡ Γ(µ+→e+γ)
Γ(µ+→e+νν)¯ <1.2×10−11 90% C.L. (9.29) Theoretically, a general analysis in terms of slepton and sneutrino mixing matrices has been given in Ref. [60]. Using the notation of that work, we consider those contributions to the µ → eγ amplitude that contain the same combinations for LFV mixing matrices as appear in the δ˜(b)µ . For simplicity, we also set (in the present analytical estimate, but not in the following numerical computation) the chargino and neutralino mixing matrices
to unity and neglect contributions that are suppressed by factors of mµ/MW. With these approximations, the combination Zν1mZν2m appears only in the first term of the chargino loop amplitudeA(c)R2 according to the notation of Ref. [60]. Setting the chargino mixing to 1 (or, equivalently, considering the pure wino contribution alone), gives
A(c)R2 ' α 8πsin2θW
1 m2eνm
¡Zν1mZν2m¢
f(c)(xm) +· · · , xm =
µmeνm mWf
¶
(9.30)
where f(c)(x) is a loop function. Analogously, the combination ZL4iZL5i appears only in the first term of the neutralino loop amplitudeA(n)L2 . This time the amplitude reads (again considering only the purebinoloop)
A(n)L2 ' α 4πcos2θW
1 m2e
Li
¡ZL4iZL5i¢
f(n)(xi) +· · · , xi = µmLei
mBe
¶
(9.31)
The resulting muon decay widths respectively read Γ(c)(µ→eγ)' α
4
α2 (8πsin2θW)2
m5µ m4νem
¡Zν1mZν2m¢2¡
f(c)(xm)¢2
(9.32) and
Γ(n)(µ→eγ)' α 4
α2 (4πcos2θW)2
m5µ m4Le
i
¡ZL4iZL5i¢2¡
f(n)(xi)¢2
(9.33) For simplicity, we consider two extremes: mνem ≈ mLei ≈mWf ≈ mBe=100, 1000 GeV
≡M˜. For either choice, we find
¡f(c)(1)¢2 '¡
f(n)(1)¢2
'0.007 (9.34)
Inserting the numerical values, and requiring that Γ(n,c) . 10−30 GeV as required by the limit (9.29), we find that
¡ZL4iZL5i¢
max≈¡
Zν1mZν2m¢
max<
10−3 for ˜M = 100 GeV 10−1 for ˜M = 1000 GeV
(9.35)
10-14 10-12 10-10 10-8 BrHΜ®eΓL
10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6
10-11 10-10 10-9 10-8 10-7 10-6 10-5
ÈDΤΜÈΤΜSM ÈgSRRÈ
BrHΜ®eΓL£1.2*10-11 Any BrHΜ®eΓL
1 2 3 4 5 6 7
∆ 10-13
10-12 10-11 10-10 10-9 10-8 10-7 10-6
10-11 10-10 10-9 10-8 10-7 10-6 10-5
ÈDΤΜÈΤΜSM ÈgSRRÈ
BrHΜ®eΓL£1.2*10-11 Any BrHΜ®eΓL
(a) (b)
Figure 9.3: A scatter plot showing|∆τµ|/τµSM(left vertical axis) and|gRRS |(right vertical axis), as functions of Br(µ→ eγ) (a) and ofδLFV (b) (smallerδLFVmeans more lepton flavor mixing; see the text for the precise definition). Filled circles represent models consistent with the current bound Br(µ→eγ)≤1.2×10−11, while empty circles denote all other models.
This implies that for superpartner masses of the order of 100 GeV, the amplitudesδ˜(b.1,b.2)µ
are suppressed by a factor10−6 relative to the naive expectations discussed above, while for 1000 GeV masses by a factor10−2. In this latter case, however, the loop functions in the amplitudesδ˜µ(b)experience a further suppression factor of order10−2. Thus, the magnitude of theδ˜(b)µ should be no larger than∼10−7.
We substantiate the previous estimates by performing a numerical scan over the pa- rameter space of theCP-conserving MSSM [117]. We do not implement any universality assumption in the slepton soft supersymmetry breaking mass sector or in the gaugino mass terms. However, in this section only, we neglect L-R mixing and consider flavor mixing be-
µ m1 m2 (M2LL)ij,(M2RR)ij tanβ 30÷10000 2÷1000 50÷1000 102÷20002 1÷60
Table 9.1: Ranges of the MSSM parameters used to generate the models shown in Fig. 9.3. Here, µis the usual higgsino mass term, whilem1,2 indicate the soft supersymmetry breaking U(1)Y and SU(2) gaugino masses. The matricesM2LLandM2RR are symmetric; hence, we scanned over 6 independent masses within the specified range. All masses are in GeV.
tween the first and second generation sleptons only. Under these assumptions, the mixing that causes non-vanishing ˜δµ(b.1,2) stems solely from off-diagonal elements in the two 2×2 slepton mass matrices M2LL and M2RR. We scan independently over all the parameters indicated in Table 9.1, within the specified ranges. For all models, we impose constraints from direct supersymmetric-particles searches at accelerators and require the lightest su- persymmetric particle (LSP) to be the lightest neutralino (see also [118] for more details).
The result of this scan is shown in Fig. 9.3. Although general models can accommodate gRRS ∼10−5, the current constraint on Br(µ→eγ) severely restricts the available parameter space and reduces the allowed upper limit ongSRRby over an order of magnitude, as shown in Fig. 9.3 (a). It is also instructive to exhibit the sensitivity ofgSRRto the degree of flavor- mixing and to show the corresponding impact of the LFV searches. To that end, we quantify the amount of lepton flavor mixing by a parameterδLFV, defined as
δLFV =|δL|+|δR|, (9.36)
where δL= log
µ 2 (M2LL)12 (M2LL)11+ (M2LL)22
¶
δR= log
µ 2 (M2RR)12 (M2RR)11+ (M2RR)22
¶
(9.37) ande.g.,(M2LL)ij is the(i, j)-th component of left-handed slepton mass matrix. For exam- ple, if|δL|is close to zero, then there is a large flavor mixing contribution from left-handed sleptons; but if |δL| is large, then this flavor mixing is suppressed. Since the amplitudes δ˜(b.1,2)µ depend on flavor mixing amongbothLH and RH sleptons, they contribute only if both δL and δR are small. (In contrast, Br(µ → eγ) survives in the presence of flavor mixing amongeitherLH or RH sleptons.) Naturally, the flavor mixing contribution togRRS is largest when δLF V is smallest, as shown in Fig. 9.3(b). We note that to obtain a large flavor mixing contribution togRRS , it is not sufficient simply to have maximal mixing (i.e.
|ZLij|= 1/√
2); in addition, one needs the absence of a degeneracy among the slepton mass eigenstates, or else the sum over mass eigenstates [e.g., sum overi andi0 in Eqn (9.24)]
will cancel. In any case, we conclude that the flavor mixing box graph contributions are too
small to be important for the interpretation of the next generation muon decay experiments, where the precision inτµis expected to be of the order of one ppm.